1. CS 188: Artificial Intelligence Fall 2008
Lecture 2: Queue-Based Search
9/2/2008
Dan Klein – UC Berkeley
Many slides from either Stuart Russell or Andrew Moore

2. Announcements Written assignments:
One mini-homework each week (1-2 problems)
We’ll drop your two lowest scores
First one posted soon!
Lab Wednesday 11am to 4pm in Soda 275
Learn Python
Come whatever times you like
Project 1.1 posted soon, too, due 9/12

3. Today Agents that Plan Ahead Search Problems Uniformed Search Methods
Depth-First Search
Breadth-First Search
Uniform-Cost Search
Heuristic Search Methods
Greedy Search
A* Search

4. Reflex Agents Reflex agents: Can a reflex agent be rational?
Choose action based on current percept and memory
May have memory or a model of the world’s current state
Do not consider the future consequences of their actions
Can a reflex agent be rational?
[demo: reflex optimal / loop ]

5. Goal Based Agents Goal-based agents: Plan ahead
Decisions based on (hypothesized) consequences of actions
Must have a model of how the world evolves in response to actions
[demo: plan fast / slow ]

6. Search Problems A search problem consists of:
A state space
A successor function
A start state and a goal test
A solution is a sequence of actions (a plan) which transforms the start state to a goal state
“N”, 1.0
“E”, 1.0

7. Search Trees A search tree:
“N”, 1.0
“E”, 1.0
A search tree:
This is a “what if” tree of plans and outcomes
Start state at the root node
Children correspond to successors
Nodes labeled with states, correspond to PLANS to those states
For most problems, can never actually build the whole tree
So, have to find ways of using only the important parts of the tree!

8. Laughably tiny search graph for a tiny search problem
State Space Graphs
There’s some big graph in which
Each state is a node
Each successor is an outgoing arc
Important: For most problems we could never actually build this graph
How many states in Pacman? S G d b p q c e h a f r
Laughably tiny search graph for a tiny search problem

9. Example: Romania

10. Another Search Tree Search: Expand out possible plans
Maintain a fringe of unexpanded plans
Try to expand as few tree nodes as possible

11. Detailed pseudocode is in the book!
General Tree Search
Important ideas:
Fringe
Expansion
Exploration strategy
Main question: which fringe nodes to explore?
Detailed pseudocode is in the book!

12. Example: Tree Search S G d b p q c e h a f r

13. State Graphs vs Search Treesd b p q c e h a f r
Each NODE in in the search tree is an entire PATH in the problem graph. S a b d p c e h f r q G
We almost always construct both on demand – and we construct as little as possible.

14. Review: Depth First SearchG d b p q c e h a f r a
Strategy: expand deepest node first
Implementation: Fringe is a LIFO stack b c e d f h p r q S a b d p c e h f r q G

15. Review: Breadth First SearchG d b p q c e h a f r
Strategy: expand shallowest node first
Implementation: Fringe is a FIFO queue
Search
Tiers S a b d p c e h f r q G

16. Search Algorithm Properties
Complete? Guaranteed to find a solution if one exists?
Optimal? Guaranteed to find the least cost path?
Time complexity?
Space complexity?
Variables: n
Number of states in the problem b
The average branching factor B
(the average number of successors) C*
Cost of least cost solution s
Depth of the shallowest solution m
Max depth of the search tree

17. DFS Infinite paths make DFS incomplete… How can we fix this? O(BLMAX)
Algorithm
Complete
Optimal
Time
Space
DFS
Depth First Search N
O(BLMAX)
O(LMAX) N N
Infinite
Infinite b
START a
GOAL
Infinite paths make DFS incomplete…
How can we fix this?

18. DFS With cycle checking, DFS is complete. When is DFS optimal?
1 node b …
b nodes
b2 nodes
m tiers
bm nodes
Algorithm
Complete
Optimal
Time
Space
DFS
w/ Path Checking Y N
O(bm+1)
O(bm)

19. BFS Algorithm Complete Optimal Time Space DFS BFS When is BFS optimal?
w/ Path Checking
BFS
When is BFS optimal? Y N
O(bm+1)
O(bm) Y N*
O(bs+1)
O(bs)
1 node b …
b nodes
s tiers
b2 nodes
bs nodes
bm nodes

20. Comparisons When will BFS outperform DFS?
When will DFS outperform BFS?

21. Iterative Deepening Algorithm Complete Optimal Time Space DFS BFS ID Y
Iterative deepening uses DFS as a subroutine:
Do a DFS which only searches for paths of length 1 or less. (DFS gives up on any path of length 2)
If “1” failed, do a DFS which only searches paths of length 2 or less.
If “2” failed, do a DFS which only searches paths of length 3 or less.
….and so on. b …
Algorithm
Complete
Optimal
Time
Space
DFS
w/ Path Checking
BFS ID Y N
O(bm+1)
O(bm) Y N*
O(bs+1)
O(bs) Y N*
O(bs+1)
O(bs)

22. Costs on Actions
START
GOAL d b p q c e h a f r 2 9 8 1 3 4 15
Notice that BFS finds the shortest path in terms of number of transitions. It does not find the least-cost path.
We will quickly cover an algorithm which does find the least-cost path.

23. Uniform Cost Search S G 2 Expand cheapest node first:b p q c e h a f r 2
Expand cheapest node first:
Fringe is a priority queue 1 8 2 2 3 9 1 8 1 1 15 S a b d p c e h f r q G 9 1 3 4 5 17 11 16 11
Cost contours 6 13 7 8 11 10

24. Priority Queue Refresher
A priority queue is a data structure in which you can insert and retrieve (key, value) pairs with the following operations:
You can promote or demote keys by resetting their priorities
Unlike a regular queue, insertions into a priority queue are not constant time, usually O(log n)
We’ll need priority queues for most cost-sensitive search methods.
pq.push(key, value)
inserts (key, value) into the queue.
pq.pop()
returns the key with the lowest value, and removes it from the queue.

25. Uniform Cost Search Algorithm Complete Optimal Time Space DFS BFS UCS
w/ Path Checking
BFS
UCS Y N
O(bm+1)
O(bm) Y N
O(bs+1)
O(bs) Y* Y
O(C* bC*/)
O(bC*/)
C*/ tiers b …
We’ll talk more about uniform cost search’s failure cases later…

26. Uniform Cost Problems Remember: explores increasing cost contours
The good: UCS is complete and optimal!
The bad:
Explores options in every “direction”
No information about goal location …
c  1
c  2
c  3
Start
Goal
[demo: ucs contours ]

27. Heuristics

28. Best First / Greedy Search
Expand the node that seems closest…
What can go wrong?

29. Best First / Greedy Search
START
GOAL d b p q c e h a f r 2 9 8 1 3 5 4 15
h=0
h=8
h=5
h=4
h=11
h=8
h=4
h=6
h=12
h=11
h=6
h=9

30. Best First / Greedy Search
A common case:
Best-first takes you straight to the (wrong) goal
Worst-case: like a badly-guided DFS in the worst case
Can explore everything
Can get stuck in loops if no cycle checking
Like DFS in completeness (finite states w/ cycle checking) b … b …

31. Search Gone Wrong?

32. Extra Work?
Failure to detect repeated states can cause exponentially more work. Why?

33. Graph Search
In BFS, for example, we shouldn’t bother expanding the circled nodes (why?) S a b d p c e h f r q G

34. Graph Search Very simple fix: never expand a state type twice
Can this wreck completeness? Why or why not?
How about optimality? Why or why not?

35. Some Hints
Graph search is almost always better than tree search (when not?)
Fringes are sometimes called “closed lists” – but don’t implement them with lists (use sets)!
Nodes are conceptually paths, but better to represent with a state, cost, and reference to parent node